-1
$\begingroup$

I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated.

(I am trying to find whether the map is area preserving? I know "the map $f:\mathbb{R}^n\to\mathbb{R}^n$ is area and orientation preserving iff the determinant of the Jacobian is $\pm1$".)

$\endgroup$
1
  • 1
    $\begingroup$ What is the definition of Jacobian? $\endgroup$ – zoidberg Dec 9 '18 at 2:39
0
$\begingroup$

The Jacobian of a multivariable function $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(u(x,y),v(x,y))$ is:

$$\textbf{J}(f(x,y)) = det\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{pmatrix}$$

where $det$ denotes the determinant of the matrix.

Calculating the partial derivatives of $u(x,y)=x+y^2$ and $v(x,y)=y+x^2$ we have that the Jacobian of your given function is:

$$det\begin{pmatrix} 1 & 2y \\ 2x & 1 \\ \end{pmatrix}=1-4xy$$

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.